[Proj] Kruger 1912 inverse formulas are faster than I claimed.

Karney, Charles ckarney at Sarnoff.com
Wed Apr 1 07:27:27 EST 2009

I have code (not part of libproj) which implements the transverse
Mercator projection by the following methods:

    * Krueger series of order 6 (but the order can be selected in [4,8]
      at compile time);
    * "exact" method (Thompson and Lee) in terms of elliptic functions.

These routines also compute the convergence and scale of the projection.
For documentation go to


The code itself (C++ classes and a utility, TransverseMercatorTest, for
testing the two implementations) is available at


The exact method is accurate to 8 nm everywhere and allows the accuracy
of approximate methods to be measured.  I find that the 6th order
Krueger method is accurate to 5 nm, within 4200 km (about 35 arc
degrees) of the central meridian.  (Obviously, I had to use a high
precision version of the exact method to make this determination.)

The errors in the Krueger method are primarily a function of distance
from the meridian (not longitude); hence there's no penalty for
countries at high latitudes.

My focus has mostly been on accuracy over speed, so I make no claims on
how fast these routines are beyond the following:

    * the exact routines are about 4.5 times slower than the 6th-order
    * the speed of the series solution is not very sensitive to its

Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300

URL: http://charles.karney.info
Tel: +1 609 734 2312
Fax: +1 609 734 2662

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